Use synthetic division to decide whether the given number is a solution of the given equation.

[tex]$-x^4-3 x^2-x-8 ; x=3$[/tex]

A. Yes
B. No

After substituting x = 3 into the polynomial and evaluating, the result is − 119 , which is not equal to 0. Therefore, x = 3 is not a solution to the equation. The correct answer is N o ​ .
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Answered by Anonymous | 2025-07-04

Substitute x = 3 into the polynomial − x 4 − 3 x 2 − x − 8 .
Evaluate the expression: − ( 3 ) 4 − 3 ( 3 ) 2 − ( 3 ) − 8 = − 81 − 27 − 3 − 8 = − 119 .
Since the result is − 119  = 0 , x = 3 is not a solution.
Therefore, the answer is N o ​ .

Explanation

Understanding the Problem We are given the polynomial − x 4 − 3 x 2 − x − 8 and asked to determine if x = 3 is a solution to the equation − x 4 − 3 x 2 − x − 8 = 0 using synthetic division. Since we are asked to use synthetic division, we can evaluate the polynomial at x = 3 and check if the result is zero. If it is, then x = 3 is a solution. If it is not, then x = 3 is not a solution.

Substituting x=3 To determine if x = 3 is a solution, we substitute x = 3 into the polynomial: − ( 3 ) 4 − 3 ( 3 ) 2 − ( 3 ) − 8

Evaluating the Expression Now, we evaluate the expression: − ( 3 ) 4 − 3 ( 3 ) 2 − ( 3 ) − 8 = − 81 − 3 ( 9 ) − 3 − 8 = − 81 − 27 − 3 − 8 = − 119

Conclusion Since the result is − 119 , which is not equal to 0, x = 3 is not a solution to the equation − x 4 − 3 x 2 − x − 8 = 0 .


Examples
Imagine you are designing a bridge and need to ensure that a certain support can withstand a specific load. You can model the load as a polynomial equation and use synthetic division to check if a particular load value will cause the support to fail. If the remainder is zero, the support will fail under that load; otherwise, it will withstand it. This helps engineers design safe and reliable structures.

Answered by GinnyAnswer | 2025-07-04